ReviewNo Access

TIME AS A QUANTUM OBSERVABLE

V. S. OLKHOVSKY

Institute for Nuclear Research of NASU, Kiev-03028, Ukraine

Search for more papers by this author

and

E. RECAMI

Facoltà di Ingegneria, Università statale di Bergamo, Bergamo, Italy

INFN-Sezione di Milano, Milan, Italy

Search for more papers by this author

https://doi.org/10.1142/S0217751X0703724XCited by:29 (Source: Crossref)

Abstract

Some new results are presented and recent developments are reviewed on the study of Time in quantum physics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the Hermitian (more precisely, maximal Hermitian, but non-self-adjoint) operator for Time which appears: (i) for particles, in ordinary nonrelativistic quantum mechanics; and (ii) for photons (i.e. in first-quantization quantum electrodynamics). In conclusion, various recent and possible future applications of the time quantum analysis for tunnelling processes, nuclear collisions and systems with (quasi)discrete energy spectra are indicated.

Keywords:

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

W. Pauli, Handbuch der Physik 5/1, ed. S. Fluegge (Springer, Berlin, 1926) p. 60. Google Scholar
W. Pauli , General Principles of Quantum Theory ( Springer , Berlin , 1980 ) . Crossref, Google Scholar
V. S. Olkhovsky and E. Recami, Lett. Nuovo Cimento (1st Series) 4, 1165 (1970), DOI: 10.1007/BF02753666. Crossref, Web of Science, Google Scholar
V. S. Olkhovsky, Ukrainskiy Fiz. Zhurnal 18, 1910 (1973). Google Scholar
V. S. Olkhovsky, E. Recami and A. Gerasimchuk, Nuovo Cimento A 22, 263 (1974), DOI: 10.1007/BF02813438. Crossref, Web of Science, ADS, Google Scholar
E. Recami, The Uncertainty Principle and Foundation of Quantum Mechanics (J. Wiley, London, 1977) pp. 21–28. Google Scholar
E. Recami, Recent Developments in Relativistic QFT and its Applications II, ed. W. Karkowski (Wroclaw University Press, 1976) pp. 256–265. Google Scholar
V. S. Olkhovsky, Sov. J. Part. Nucl. 15, 130 (1984). Web of Science, Google Scholar
V. S. Olkhovsky, Nukleonika 35, 99 (1990). Google Scholar
V. S. Olkhovsky, Mysteries, Puzzles and Paradoxes in Quantum Mechanics, ed. R. Bonifacio (AIP, 1998) pp. 272–276. Google Scholar
A. S. Holevo, Rep. Math. Phys. 13, 379 (1978), DOI: 10.1016/0034-4877(78)90065-4. Crossref, ADS, Google Scholar
A. S. Holevo , Probabilistic and Statistical Aspects of Quantum Theory ( North-Holland , Amsterdam , 1982 ) . Google Scholar
M. D. Srinivas and R. Vijayalakshmi, Pramana J. Phys. 16, 173 (1981), DOI: 10.1007/BF02848181. Crossref, Web of Science, ADS, Google Scholar
P. Busch, M. Grabowski and P. J. Lahti, Phys. Lett. A 191, 357 (1994), DOI: 10.1016/0375-9601(94)90785-4. Crossref, Web of Science, ADS, Google Scholar
D. H. Kobe and V. C. Aguilera-Navarro, Phys. Rev. A 50, 933 (1994), DOI: 10.1103/PhysRevA.50.933. Crossref, Web of Science, ADS, Google Scholar
D. H. Kobeet al., Phys. Rev. A 64, 022104 (2001), DOI: 10.1103/PhysRevA.64.022104. Crossref, Web of Science, Google Scholar
P. Blanchard and A. Jadczyk, Helv. Phys. Acta 69, 613 (1996). Google Scholar
N. Grot, C. Rovelli and R. S. Tate, Phys. Rev. A 54, 4676 (1996), DOI: 10.1103/PhysRevA.54.4676. Crossref, Web of Science, ADS, Google Scholar
J. Leòn, J. Phys. A 30, 4791 (1997). Crossref, ADS, Google Scholar
Y. Aharonovet al., Phys. Rev. A 57, 4130 (1998), DOI: 10.1103/PhysRevA.57.4130. Crossref, Web of Science, ADS, Google Scholar
H. Atmanspacher and A. Amann, Int. J. Theor. Phys. 37, 629 (1998), DOI: 10.1023/A:1026604126629. Crossref, Web of Science, Google Scholar
V. S. Olkhovsky and E. Recami, Phys. Rep. 214, 339 (1992), DOI: 10.1016/0370-1573(92)90015-R. Crossref, Web of Science, ADS, Google Scholar
V. S. Olkhovskyet al., J. de Phys.-I (France) 5, 1351 (1995), DOI: 10.1051/jp1:1995202. Crossref, Google Scholar
V. S. Olkhovsky, E. Recami and J. Jakiel, Phys. Rep. 398, 133 (2004), DOI: 10.1016/j.physrep.2004.06.001. Crossref, Web of Science, ADS, Google Scholar
V. S. Olkhovsky and A. Agresti, Proc. Adriatico Research Conf. on Tunnelling and its Implications (World Scientific, Singapore, 1997) pp. 327–355. Google Scholar
M. Toller, Phys. Rev. A 59, 960 (1999), DOI: 10.1103/PhysRevA.59.960. Crossref, Web of Science, ADS, Google Scholar
V. Delgado, Phys. Rev. A 59, 1010 (1999), DOI: 10.1103/PhysRevA.59.1010. Crossref, Web of Science, Google Scholar
R. Giannitrapani, Int. J. Theor. Phys. 36, 1575 (1997), DOI: 10.1007/BF02435757. Crossref, Web of Science, Google Scholar
J. Muga, J. Papao and C. Leavens, Phys. Lett. A 253, 21 (1999), DOI: 10.1016/S0375-9601(99)00020-1. Crossref, Web of Science, ADS, Google Scholar
J. Mugaet al., Phys. Rev. A 66, 042115 (2002), DOI: 10.1103/PhysRevA.66.042115. Crossref, Web of Science, Google Scholar
P. Kochànski and K. Wòdkievicz, Phys. Rev. A 60, 2689 (1999). Crossref, Web of Science, Google Scholar
M. A. Naimark, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 4, 277 (1940). Google Scholar
V. S. Olkhovsky and E. Recami, Nuovo Cimento A 53, 610 (1968), DOI: 10.1007/BF02721712. Crossref, Web of Science, ADS, Google Scholar
V. S. Olkhovsky and E. Recami, Nuovo Cimento A 63, 814 (1969), DOI: 10.1007/BF02760741. Crossref, Web of Science, ADS, Google Scholar
V. S. Olkhovsky and E. Recami, Lett. Nuovo Cimento 4, 1165 (1970), DOI: 10.1007/BF02753666. Crossref, Web of Science, Google Scholar
A. J. Kàlnay, Boletin del IMAF (Còrdoba) 2, 11 (1966). Google Scholar
A. J. Kàlnay and B. P. Toledo, Nuovo Cimento A 48, 997 (1967). Crossref, Web of Science, ADS, Google Scholar
J. A. Gallardoet al., Nuovo Cimento A 48, 1008 (1967), DOI: 10.1007/BF02721624. Crossref, Web of Science, Google Scholar
J. A. Gallardoet al., Nuovo Cimento A 49, 393 (1967), DOI: 10.1007/BF02820318. Crossref, Web of Science, ADS, Google Scholar
J. A. Gallardo, A. J. Kàlnay and S. H. Risenberg, Phys. Rev. 158, 1484 (1967), DOI: 10.1103/PhysRev.158.1484. Crossref, Web of Science, ADS, Google Scholar
E. Recami, Atti Accad. Naz. Lincei (Roma) 49, 77 (1970). Google Scholar
M. Baldo and E. Recami, Lett. Nuovo Cimento 2, 613 (1969), DOI: 10.1007/BF02755821. Web of Science, Google Scholar
N. I. Akhiezer and I. M. Glazman , The Theory of Linear Operators in Hilbert Space ( Pitman , Boston, Massachusetts , 1981 ) . Google Scholar
J. von Neumann , Mathematischen Grundlagen del Quantum Mechanik ( Hizzel , Leipzig , 1932 ) . Google Scholar
M. H. Stone, Proc. Nat. Acad. Sci. USA 16, N1 (1930), DOI: 10.1073/pnas.16.2.172. Google Scholar
D. ter Haar , Elements of Hamiltonian Mechanics ( Pergamon Press , Oxford , 1971 ) . Google Scholar
V. S. Olkhovsky and E. Recami, Nuovo Cimento A 53, 610 (1968), DOI: 10.1007/BF02721712. Crossref, Web of Science, ADS, Google Scholar
E. Recami, W. A. Rodrigues and P. Smrz, Hadronic J. 6, 1773 (1983). Google Scholar
D. M. Rosenbaum, J. Math. Phys. 10, 1127 (1969), DOI: 10.1063/1.1664949. Crossref, Web of Science, ADS, Google Scholar
S. Schweber , An Introduction to Relativistic Quantum Field Theory ( Row, Peterson and Co. , 1961 ) . Google Scholar
A. I. Akhiezer and V. B. Berestezky , Quantum Electrodynamics ( Fizmatgiz , Moscow , 1959 ) . Google Scholar
D. Judge and J. L. Levis, Phys. Lett. 5, 190 (1963). Crossref, Web of Science, ADS, Google Scholar
P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968), DOI: 10.1103/RevModPhys.40.411. Crossref, Web of Science, ADS, Google Scholar
L. I. Schiff , Quantum Mechanics ( McGraw-Hill , 1955 ) . Google Scholar
A. S. Davydov , Quantum Mechanics ( Nauka , Moscow , 1971 ) . Google Scholar
A. S. Davydov , Quantum Mechanics ( Pergamon , Oxford , 1982 ) . Google Scholar
V. S. Olkhovsky and N. L. Doroshko, Europhys. Lett. 18, 483 (1992), DOI: 10.1209/0295-5075/18/6/002. Crossref, ADS, Google Scholar
A. D'Arrigoet al., Int. J. Mod. Phys. E 2, 809 (1993). Link, ADS, Google Scholar
N. V. Ereminet al., Mod. Phys. Lett. A 9, 2849 (1994), DOI: 10.1142/S0217732394002690. Link, Web of Science, Google Scholar
V. S. Olkhovsky, M. E. Dolinska and S. A. Omelchenko, Cent. Eur. J. Phys. 4, 1 (2006), DOI: 10.2478/s11534-006-0008-z. Google Scholar
M. Abolhasani and M. Golshani, Phys. Rev. A 62, 012106 (2000), DOI: 10.1103/PhysRevA.62.012106. Crossref, Web of Science, Google Scholar
V. S. Olkhovsky, V. Petrillo and A. K. Zaichenko, Phys. Rev. A 70, 034103 (2004), DOI: 10.1103/PhysRevA.70.034103. Crossref, Web of Science, Google Scholar
N. S. Krylov and V. A. Fock, Sov. J. ZHETF 17, 93 (1947). Google Scholar
V. A. Fock, Sov. J. ZHETF 42, 1135 (1962). Google Scholar
Y. Aharonov and D. Bohm, Phys. Rev. 122, 1649 (1961), DOI: 10.1103/PhysRev.122.1649. Crossref, Web of Science, ADS, Google Scholar
Y. Aharonov and D. Bohm, Phys. Rev. B 134, 1417 (1964). Crossref, Web of Science, Google Scholar